There is no such thing as the "Jewish race" - that's an invention of so-called scientific racists of the late 19th and early 20th century. (I should mention that all concepts of "race" are socially constructed.) And in this view, race was all that mattered. Religion was of little importance when it came to how Europeans wanted to deal with their Jewish neighbors.

That being said, the Nazis did not differentiate between people who practiced the Jewish religion and those the Nazis considered to be racially Jewish (which included many people who were not religiously Jewish, such as converts and their children). As far as the Nazis were concerned, both were equally Jewish and therefore, equally deserving of extermination.

EDIT: Also, the Nazis used many method of determining one's Jewish identity. Membership in Jewish organizations or associations, participation in the Jewish religion, or family ancestry were all indicators of one's Jewishness. One's confession could also be obtained through some census records. It wasn't difficult to determine who was Jewish through the records already in place. You also have to remember that in the early days of the Nazi Parry's time in power, many people, including Jews, did not believe that the Nazis would actually do what they had said (destroy Europe's Jewish population). They looked at the Nazis as yet another manifestation of European anti-semitism that had flourished for centuries. Many Jews thought they could outlast the problem as they had outlasted previous bouts of anti-antisemitism in previous decades. Many people could not fathom that Germany, the land of Goethe and Schiller, the land of Weimar democracy and liberalism, a land otherwise very friendly to Jewish citizens in the 20s, would devolve into the genocidal regime the Nazis instituted. So in those early days, there wasn't necessarily the need to hide one's Jewishness, as it wasn't' believed to be, quite literally, a marker for future execution.